7 research outputs found
Memory-Constrained Algorithms for Simple Polygons
A constant-workspace algorithm has read-only access to an input array and may
use only O(1) additional words of bits, where is the size of
the input. We assume that a simple -gon is given by the ordered sequence of
its vertices. We show that we can find a triangulation of a plane straight-line
graph in time. We also consider preprocessing a simple polygon for
shortest path queries when the space constraint is relaxed to allow words
of working space. After a preprocessing of time, we are able to solve
shortest path queries between any two points inside the polygon in
time.Comment: Preprint appeared in EuroCG 201
Space-Time Trade-offs for Stack-Based Algorithms
In memory-constrained algorithms we have read-only access to the input, and
the number of additional variables is limited. In this paper we introduce the
compressed stack technique, a method that allows to transform algorithms whose
space bottleneck is a stack into memory-constrained algorithms. Given an
algorithm \alg\ that runs in O(n) time using variables, we can
modify it so that it runs in time using a workspace of O(s)
variables (for any ) or time using variables (for any ). We also show how the technique
can be applied to solve various geometric problems, namely computing the convex
hull of a simple polygon, a triangulation of a monotone polygon, the shortest
path between two points inside a monotone polygon, 1-dimensional pyramid
approximation of a 1-dimensional vector, and the visibility profile of a point
inside a simple polygon. Our approach exceeds or matches the best-known results
for these problems in constant-workspace models (when they exist), and gives
the first trade-off between the size of the workspace and running time. To the
best of our knowledge, this is the first general framework for obtaining
memory-constrained algorithms
Solving Geometric Problems in Space-Conscious Models
When dealing with massive data sets, standard algorithms may
easily ``run out of memory''. In this thesis, we design efficient
algorithms in space-conscious models. In particular, in-place
algorithms, multi-pass algorithms, read-only algorithms, and
stream-sort algorithms are studied, and the focus is on
fundamental geometric problems, such as 2D convex hulls, 3D convex
hulls, Voronoi diagrams and nearest neighbor queries, Klee's
measure problem, and low-dimensional linear programming.
In-place algorithms only use O(1) extra space besides the input
array. We present a data structure for 2D nearest neighbor queries
and algorithms for Klee's measure problem in this model.
Algorithms in the multi-pass model only make read-only sequential
access to the input, and use sublinear working space and small
(usually a constant) number of passes on the input. We present
algorithms and lower bounds for many problems, including
low-dimensional linear programming and convex hulls, in this
model.
Algorithms in the read-only model only make read-only random
access to the input array, and use sublinear working space. We
present algorithms for Klee's measure problem and 2D convex hulls
in this model.
Algorithms in the stream-sort model use sorting as a primitive
operation. Each pass can either sort the data or make sequential
access to the data. As in the multi-pass model, these algorithms
can only use sublinear working space and a small (usually a
constant) number of passes on the data. We present algorithms for
constructing convex hulls and polygon triangulation in this model